Tuesday, April 29, 2008

Next week I'm attending a symposium at the Royal Society on the origin of time and the universe, in honour of Michael Heller's 2008 Templeton Prize. There'll be talks by Michael himself, John Barrow, Andreas Doring, Paul Tod, and Shahn Majid. Majid proposed some years ago that the mathematical concept of self-duality can be used to provide an ultimate explanation of the universe in terms of the universe itself; a type of self-explanation. Michael Heller wrote a lovely paper on Majid's ideas, (Algebraic self-daulity as the 'ultimate explanation', Foundations of Science, Vol. 9, pp369-385) in 2004, and I've been reviewing this ahead of next week's symposium.

Majid illustrates his idea with the notion of a self-dual bicrossproduct Hopf algebra, (although he doesn't believe that this specific mathematical structure is a candidate for a theory of everything). To understand what this is, we first need to understand what a Hopf algebra is.

An algebra is a structure which possesses both the addition operation '+' of a vector space over a number field K, and a product operation '⋅'. An algebra is unital if there is a mapping η:K → H. A bialgebra possesses both the structure of an associative unital algebra (H,+,⋅,η,K), and the structure of a coalgebra (H,+,Δ,ε,K), where Δ:H → H × H is the coproduct, and ε:H → K is the counit. Given an algebra or coalgebra A, the dual A* is defined to be set of linear functionals on φ: A → K. The dual of an algebra is a natural colagebra, and the dual of a coalgebra is a natural algebra, hence a bialgebra possesses within itself the dual of both its algebra and coalgebra. A Hopf algebra H is a bialgebra equipped with an special map S:H → H called the antipode map.

Now, suppose that H and A are both Hopf algebras, and suppose that there is a left-action of H on A, and a right-coaction on A on H. This means that there is a mapping α: H × A → A, and a mapping β: H → H × A, respectively. One can then form something called the cross-product H ♦ A, which will also be a Hopf algebra. This cross-product has a dual cross-product, and the combination of the two cross-products provides the structure of a bicrossproduct Hopf algebra.

The dual of this bicrossproduct gives another bicrossproduct, but with H exchanged for the dual A*, and with A exchanged for H*. Thus, if H is isomorphic to A* (and A isomorphic to H*), then one has a self-dual bicrossproduct Hopf algebra.

Majid's proposal is that such a structure is self-explaining, in the sense, I think, that the A-part explains the H-part, the H-part explains the A-part, the whole thing H ♦ A is explained by either H or A, and either H or A is explained by H ♦ A. It's a type of Yin-Yang idea.

Now, this is fascinating stuff, but I think the overall objective of seeking a self-explaining universe, is doomed to failure. Suppose, for the sake of illustration, that our universe were a self-dual bicrossproduct Hopf algebra. This would not eliminate the question of contingency. One could still ask: why does this particular self-dual bicrossproduct Hopf algebra physically exist, and not one of the many other self-dual bicrossproduct Hopf algebras? The question is only eliminated if one proposes that our universe is an object in a mathematical category containing only one object. Even then, however, the problem of contingency is merely shifted, and one can pose the question: why is our universe an object in this mathematical category rather than an object in one of the many other mathematical categories? 'Why is the universe not a topos, or a highly structured pseudo-Riemannian manifold?' one could ask.

Ultimately, I think, one has to adopt Max Tegmark's proposal that all mathematical structures physically exist. There is, one can propose, only one type of existence: absence of contradiction. What we deem the physical universe is simply one particular mathematical structure, which exists by virtue of being free from contradiction. All other mathematical structures exist, but only those containing cognitive self-aware substructures possess observers capable of asking what the mathematical structure of their universe is.

Monday, April 14, 2008

No, really. At this time of year, this otherwise moribund dual carriageway cuts a path between spectacular efflorescent banks of buttercup-yellow gorse.

And this, I propose, should be taken as a precedent. Far too much of our motorways are lined by the chlorophyllically-dominated greens of various passe shrubs and trees. As a first step towards a more colourful motorway experience, the M4 around Swindon should be lined by swathes of pyrophoric tulips; the M1 around Birmingham should be lined by roses; and the M25 in Kent should be surrounded by purple rhododendrons.

Friday, April 11, 2008

If you turned onto Virgin Radio for more than half-an-hour or so in 1997 or 1998, the chances are that you would've heard Don't Speak, sung by Gwen Stefani when she was in the band No Doubt. One reason for this is that Virgin Radio never had more than 5 songs on their playlist. The other is that it's a genius pop-song.

Despite being about the break-up of a relationship, it's got what I believe music journalists call a brilliant 'hook', and Stefani's voice melds perfectly with the melody. It's simply one of the best pop songs you could hope to hear, yet it's often neglected in Top-100 compilations. Perhaps now that it's over that crucial 10-year horizon, it will acquire some nostalgic value, and begin to be recognised as a pop classic.

Sunday, April 06, 2008

Appleyard is interested in putting such work into an overall cultural context. Science, however, can also put human culture into context, and those seeking the most sophisticated field of view will be rewarded by flipping back and forth between the two perspectives.

Kaku is one of those individuals who has forged an international media career by making wildly exaggerated, and often inaccuarate claims, about the content of modern physics. For a more reliable guide to the science, John Barrow's eclectic 1998 book, Impossibility: The Limits of Science and the Science of Limits, should be consulted. It can be freely downloaded here.

Last night I travelled via about 400 sets of traffic lights along the A4 to the Oracle Centre, in Reading, to see The Orphanage.

Almost every set of traffic lights was at red when I reached it, and there was almost no transverse traffic at any of those traffic-lighted junctions. The clear purpose of the traffic lights and their Saturday-evening timing is to break up the traffic-flow along the A4 into bunches. Before many of the traffic lights, I'd be travelling along serenely, with nothing close in front, or behind, but after waiting for an interminable length of time at a number of the junctions, there would then be an impatient little pack of cars, which would jostle and jockey for position as two-lanes funelled into one on the other side of the lights.

Upon reaching the Reading Vue Cinema, we tried to take our seats, only to find that a smartly dressed man, in late middle-age, and with a mysterious foreign accent, was sitting in one of the seats, and refusing to budge.

"'Scuse me, what's your seat number?"

"No"

"I think this is our seat!"

"I don't care."

Eventually he shifted, but to an adjacent seat which also wasn't his. Presently, the occupant of that seat arrived, but this time the mysterious aristocrat totally refused to move, even when the usher came to assist.

"I'm not moving when the film's started!"

"The film hasn't started Sir."

Eventually he moved, and this time to the seat specified on his ticket.

Half-way through the film, a group of lads at the front appeared to throw a drink over two girls in front of them. The girls got up and left, the guys laughing raucously. A security man arrived, and asked them to leave. They refused. The security man said he'd call the police. The lads stayed where they were. Eventually, two luminescently-bibbed policemen arrived, and rather than immeditaely removing the lads from the cinema, appeared to argue with them for 5 to 10 minutes. Eventually, the lads were compelled to get up and leave, one of them with a hood cloaking his features in a manner which, according to David Cameron, is supposed to evoke our sympathy more than anything else.

And the film? Brilliant, without question. It's everything you've heard about it, both scary and moving. By the end, your nerves are absolutely on edge. In more ways than one...

Saturday, April 05, 2008

There are many people I've met and known in my life, who I expect I will never meet again. Placed in its full context, this is an odd thought.

Those people I once knew are now just specks in the rear-view mirror of my mind's eye, as they, and I, hurtle by different paths to personal oblivion. We met once, but we are now separate, and in a finite time, they, and I, will cease to exist, and perhaps an infinite time will then pass during which our meeting will never be repeated. Viewed in that context, those transient meetings and acquaintances seem remarkably poignant.